Abelian Varieties

A belian Varieties has been out of print for a while. Since it was written, the subject has made some great advances, and Mumfords book giving a scheme theoretic treatment has appeared (D. Mum- ford, Abelian Varieties, Tata Lecture Notes, Oxford University Press, London, 1970). However, some topics covered in my book were not covered in Mumfords; for instance, the construction of the Picard variety, the Albanese variety, some formulas concern- ing numerical questions, the reciprocity law for correspondences and its application to Kummer theory, Chows theory for the K/k-trace and image, and others. Several people have told me they still found a number of sections of my book useful. There- fore I thank Springer-Verlag for the opportunity to keep the book in print. S. LANG v FOREWORD Pour des simplifications plus subs tan- tielles, Ie developpement futur de la geometrie algebrique ne saurait manquer sans do ute d en faire apparaitre. It is with considerable pleasure that we have seen in recent years the simplifications expected by Weil realize themselves, and it has seemed timely to incorporate them into a new book. We treat exclusively abelian varieties, and do not pretend to write a treatise on algebraic groups. Hence we have summarized in a first chapter all the general results on algebraic groups that are used in the sequel. They are all foundational results.